Quartismic family
The quartisma or Saquinlu-azo comma is a comma with a ratio of 117440512/117406179 and a monzo of [24 -6 0 1 -5⟩. It is an unnoticeable comma of the 11-limit- specifically one of the the 2.9.7.11 subgroup- with a value of approximately 0.50619 cents. The quartisma is significant on account of it being the difference between a stack of five 33/32 quartertones and one 7/6 subminor third in Just Intonation. Despite that fact that the quartisma is an unnoticeable comma in JI, a number of reasonably well known EDOs (such as 17edo, 26edo and 34edo) actually fail to temper it out. In fact, there are even some EDOs such as 23edo and 70edo that seem to temper out the comma when one merely examines the patent vals for 33/32 and 7/6, yet, upon closer examination, actually fail to temper out the comma, as these calculations prove. Examples of edos that actually do temper out the quartisma are 22edo, 24edo, 46edo, 68edo, 90edo, 91edo, 92edo, 159edo, and 3125edo.
The quartismic temperament or Saquinlu-azo temperament is the temperament that tempers out this comma. This page will also list various derived temperaments that temper out this comma and thus belong in the quartismic family.
Quartismic
Comma: 117440512/117406179
No-five POTE generators: ~3/2 = 701.9826, ~33/32 = 53.3748
No-five mapping generator:
No-five Map: [<1 0 1 5|, <0 1 1 -1|, <0 0 5 1|]
No-five EDOs: 21, 22, 24, 43, 46, 89, 135, 270, 359, 494, 629, 653, 742, 877, 1012, 1236, 1506, 2159, 2248, 2383, 2518, 7419
Badness:
The following scale tree has been found:
The following rank-2 quartismic temperament MOS scales have been found:
Full 11-limit extensions
Among quartismic temperaments, there are several options for 5-limit representation depending which among the various 5-limit commas is tempered out. Adding the schisma to the list of tempered-out commas results in some form of Altierran temperament. Adding the meantone comma results in some form of Meanquarter temperament. Adding the magic comma results in some form of Coin temperament. Adding the kleisma results in some form of Kleirtismic temperament- the "kleir-" in "Kleirtismic" is pronounced the same as "Clair". Adding the tetracot comma results in some form of Doublefour temperament. Other possible extensions are listed here.
Shrutar extension
This is the 22&46 temperament. See Shrutar.
Escapade extension
This is the 22&43 temperament. See Escapade.
Godzilla extension
This is the 24&43 temperament. See Godzilla.
Altierran
The Altierran clan is the temperament clan consisting of those temperaments in which both the schisma and the quartisma are tempered out.
Commas: 32805/32768, 117440512/117406179
POTE generators: 701.7299, 53.3889
Mapping generators: 2/1, 3/2, 33/32
Map: [<1 0 15 1 5|, <0 1 -8 1 -1|, <0 0 0 5 1|]
Badness:
13-limit
Commas: 10985/10976, 32805/32768, 117440512/117406179
POTE generators: 166.0628, 53.4151
Mapping generators: 2/1, 11/10, 33/32
Map: [<1 2 -1 3 3 5|, <0 -3 24 -3 3 -11|, <0 0 0 5 1 5|]
EDOs:
Badness:
17-limit
Commas:
POTE generator:
Mapping generator:
Map:
EDOs:
Badness:
Meanquarter
The Meanquarter clan is the temperament clan consisting of those temperaments in which both the meantone comma and the quartisma are tempered out.
Commas: 81/80, 117440512/117406179
POTE generators: 697.3325, 54.1064
Mapping generators: 3/2, 33/32
Map: [<1 0 -4 1 5|, <0 1 4 1 -1|, <0 0 5 1|]
Badness:
Coin
The Coin clan is the temperament clan consisting of those temperaments in which both the magic comma and the quartisma are tempered out.
Commas: 3125/3072, 117440512/117406179
POTE generators: 380.3623, 433.3120
Mapping generators: 5/4, 9/7
Map: [<1 0 2 1 5|, <0 5 1 0 -6|, <0 0 0 5 1|]
Badness:
Kleirtismic
The Kleirtismic clan is the temperament clan consisting of those temperaments in which both the kleisma and the quartisma are tempered out.
Commas: 15625/15552, 117440512/117406179
POTE generators: 317.0291, 370.2940
Mapping generators: 6/5, (16/13?)
Map: [<1 0 1 1 5|, <0 6 5 1 -7|, <0 0 0 5 1|]
Badness:
Doublefour
The Doublefour clan is the temperament clan consisting of those temperaments in which both the tetracot comma and the quartisma are tempered out.
Commas: 20000/19683, 117440512/117406179
POTE generators: 175.9566, 52.9708
Mapping generators: 10/9, 33/32
Map: [<1 1 1 2 4|, <0 4 9 4 -4|, <0 0 0 5 1|]
EDOs: 48d, 68, 89c
Badness: